What parameter is best to increase - time or temperature?

When you reheat something, the heat from the oven is usually just penetrating the outside layers of the food. Conduction within the food is what carries the heat into the center. To demonstrate this, you can try putting something cold (not frozen), and thick in the microwave on high for about 30-45 seconds. Take it out, then cut it in half. The outside 1/4-1/2 inch will have warmed up while the core will remain very cold.

What this means is that in order to get the center hotter, usually you need more time, not more heat. While more heat will accomplish it (more heat conduction because of a greater heat gradient), it will also overcook the outer layers and leave them dry or burnt. Since you want to retain moisture, increase the time, not the temperature.

You could actually try decreasing the temperature and increasing the time. Trying it at 350 degrees for an hour may actually work better than 400 for 45-50 minutes. It will cook the outer layers more slowly, so you lose less moisture, AND give the heat more time to penetrate into the center of the dish.

This is going to be true for pretty much any frozen/chilled food, not just eggplant, and in any cooking method (oven, microwave, sous vide, etc.). Note that frozen food in the microwave presents its own challenge, because ice is not a good absorber of microwaves, but water is quite good, so you can end up with part of the dish being really hot and part of it being still frozen, depending on your microwave's hot spots.

Time is definitely the better option since cranking up the temperature too high can result in a cooked (or even burnt) outside with a cold or frozen inside. Once it's cooked through you can bring up the temperature for a little last-minute browning.

It depends, usually longer is better to get heat into the middle unless you need to develop a better crust, in which case more heat may be called for.

If longer is developing enough or more crust than desired, then lower the temp and lengthen the time some more.

Yes.

If it's a breaded cutlet not yet coated in sauce :

• Start it on lower heat to make sure it's thawed completely and warmed up (so increase the time), then crank the temperature up to crisp up the coating.

• Use a broiler for the second phase, if you have one, but make sure to keep an eye on it as things go from golden to burnt rather quickly under a broiler ... and you'll have to flip it to crisp both sides. As it's a toaster oven specifically, you could also try toasting it once it's thawed.

If it's already sauced ... then it's like Aaronut said. I'd probably favor a longer time over higher temperature, personally.

Also be aware that toaster ovens are much more likely to be inaccurate, and they're not particularly well insulated, so the ambient air greatly affects them. (ie, if it's cold in the room, it's going to hold an average lower temperature than in a hot room) If your toaster oven is chronicly slow, you might want to try checking it with an oven thermometer, or just turn it up a few degrees when using it.